Almost every path structure is not variational

Abstract

Given a smooth family of unparameterized curves such that through every point in every direction there passes exactly one curve, does there exist a Lagrangian with extremals being precisely this family? It is known that in dimension 2 the answer is positive. In dimension 3, it follows from the work of Douglas that the answer is, in general, negative. We generalise this result to all higher dimensions and show that the answer is actually negative for almost every such a family of curves, also known as path structure or path geometry. On the other hand, we consider path geometries possessing infinitesimal symmetries and show that path and projective structures with submaximal symmetry dimensions are variational. Note that the projective structure with the submaximal symmetry algebra, the so-called Egorov structure, is not pseudo-Riemannian metrizable; we show that it is metrizable in the class of Kropina pseudo-metrics and explicitly construct the corresponding Kropina Lagrangian.